Permutations

This lesson will establish some notations and formulas which will be frequently used in problems related to counting.

Factorial

The factorial of a natural number n is the product of all natural numbers from 1 to n, that is, the product 1 x 2 x 3 x … x n. This product is denoted as n! (i.e. the number followed by an exclamation mark).

Permutations

The term permutations is used to indicate ordered arrangements of objects. For example, the permutations of the three letters Q, W and E (in a row) are QWE, QEW, WEQ, WQE, EWQ and EQW.

The number of such permutations will be 3 x 2 x 1 = 3! = 6 (We’ve already seen the method of calculation in a previous lesson)

Similarly if we had four objects to be arranged in a row, for example, forming 4-digit numbers (without repetition) using 4, 6, 7, and 9, the number of permutations will be 4 x 3 x 2 x 1 or 4!

This number permutations of objects, taken all at a time (without repetition) is denoted as \( {}^{n}P_n \) or P(n, n). As you can see, this number comes out to be n! (n-factorial)

In case we want the number of 4-digit numbers (without repetition of digits) using the digits 1 to 9, this will be 9 x 8 x 7 x 6.

This number is same as the number of permutations of 9 objects, taken 4 at a time. This is denoted as \( {}^{9}P_4 \) or P(9, 4), which equals 9 x 8 x 7 x 6. The number can be expressed using factorials. If we multiply and divide it by 5 x 4 x 3 x 2 x 1, we get P(9, 4) = (9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1)/(5 x 4 x 3 x 2 x 1) = 9!/5! or 9!/(9 – 4)!. Neat.

We can arrive at a formula now. If we have n different objects, and we have to arrange them in a row, taking r objects at a time, then the number of ways obtained will be n x (n – 1) x (n – 2) x … x (n – r + 1).

This will be equal to n!/(n – r)!, obtained by multiplying and dividing the expression by (n – r)!

The final expression obtained is denoted by \( {}^{n}P_r \) or P(n, r). That is \( {}^{n}P_r \) = n!/(n – r)!

Lesson Summary

The factorial of a natural number n, denoted by n! is equal to 1 x 2 x 3 x … x n. Zero factorial or 0! is defined as equal to 1.

The number of permutations or arrangements of n different objects in a row, taken r at a time is denoted by \( {}^{n}P_r \) which equals n!/(n – r)!

Well, that’s it. Things will get better when you go through examples, which I’ll cover next. See you there.